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Microtonal music
Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. In other words, a microtone may be thought of as a note that falls between the keys of a piano tuned in equal temperament. Terminology Microtone accidentals residing outside the Western semitone: quarter tone sharp, sharp, three quarter tones sharp; quarter tone flat, flat, (two variants of) three quarter tones flat]] Microtonal music can refer to any music containing microtones. The words "microtone" and "microtonal" were coined before 1912 by Maud MacCarthy Mann in order to avoid the misnomer "quarter tone" when speaking of the srutis of Indian music . Prior to this time the term "quarter tone" was used, confusingly, not only for an interval actually half the size of a semitone, but also for all intervals (considerably) smaller than a semitone ( ; ). It may have been even slightly earlier, perhaps as early as 1895, that the Mexican composer Julián Carrillo, writing in Spanish or French, coined the terms microtono/''micro-ton'' and microtonalismo/''micro-tonalité'' . In French, the usual term is the somewhat more self-explanatory micro-intervalle, and French sources give the equivalent German and English terms as Mikrointervall (or Kleinintervall) and micro interval (or microtone), respectively ( ; ; ; . "Microinterval" is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers ( ; ; ). In English, the two terms "microtone" and "microinterval" are synonymous . The English analogue of the related French term, micro-intervalité, however, is rare or nonexistent, normally being translated as "microtonality"; in French, the terms micro-ton, microtonal (or micro-tonal), and microtonalité are also sometimes used, occasionally mixed in the same passage with micro-intervale and micro-intervalité ( ; ; ). Ezra Sims, in the article "Microtone" in the second edition of the Harvard Dictionary of Music defines "microtone" as "an interval smaller than a semitone", which corresponds with Aristoxenus's use of the term diesis . However, the unsigned article "Comma, Schisma" in the same reference source calls comma, schisma and diaschisma "microintervals" but not "microtones" , and in the fourth edition of the same reference (which retains Sims's article on "Microtone") a new "Comma, Schisma" article by André Barbera calls them simply "intervals" . In the second edition of The New Grove Dictionary of Music and Musicians, Paul Griffiths, Mark Lindley, and Ioannis Zannos define "microtone" as a musical rather than an acoustical entity: "any musical interval or difference of pitch distinctly smaller than a semitone", including "the tiny enharmonic melodic intervals of ancient Greece, the several divisions of the octave into more than 12 parts, and various discrepancies among the intervals of just intonation or between a sharp and its enharmonically paired flat in various forms of mean-tone temperament", as well as the Indian sruti, and small intervals used in Byzantine chant, Arabic music theory from the 10th century onward, and similarly for Persian traditional music and Turkish music and various other Near Eastern musical traditions ( ), but do not actually name the "mathematical" terms schisma, comma, and diaschisma. "Microtone" is also sometimes used to refer to individual notes, "microtonal pitches" added to and distinct from the familiar twelve notes of the chromatic scale , as "enharmonic microtones" , for example. In English the word "microtonality" is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the so-called "blues scales" . It was used still earlier by W. McNaught with reference to developments in "modernism" in a 1939 record review of the Columbia History of Music, Vol. 5 . In German the term Mikrotonalität came into use at least by 1958 ( ; ), though "Mikrointervall" is still common today in contexts where very small intervals of early European tradition (diesis, comma etc.) are described, as e.g. in the new Geschichte der Musiktheorie while "Mikroton" seems to prevail in discussions of the avant-garde music and music of Eastern traditions. The term "microinterval" is used alongside "microtone" by American musicologist Margo Schulter in her articles on medieval music ( ; ). Microtonal The term "microtonal music" usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Traditional Indian systems of 22 śruti; Indonesian gamelan music; Thai, Burmese, and African music, and music using just intonation, meantone temperament or other alternative tunings may be considered microtonal ( ; ). Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues and jazz . Many microtonal equal divisions of the octave have been proposed, usually (but not always) in order to achieve approximation to the intervals of just intonation ( ; ). Terminology other than "microtonal" has been used or proposed by some theorists and composers. In 1914, A. H. Fox Strangways objected that "'heterotone' would be a better name for śruti than the usual translation 'microtone'" . Modern Indian researchers yet write: "microtonal intervals called shrutis" . In Germany, Austria, and Czechoslovakia in the 1910s and 1920s the usual term continued to be Viertelton-Musik (quarter-tone music; }}), and the type of intervallic structure found in such music was called the Vierteltonsystem ( , ), which was (in the mentioned region) regarded as the main term for referring to music with microintervals, though as early as 1908 Georg Capellan had qualified his use of "quarter tone" with the alternative term "Bruchtonstufen (Viertel- und Dritteltöne)" (fractional degrees (quarter and third-tones)) . Despite the inclusion of other fractions of a whole tone, this music continued to be described under the heading "Vierteltonmusik" until at least the 1990s, for example in the twelfth edition of the Riemann Musiklexikon , and in the second edition of the popular Brockhaus Riemann Musiklexikon . Ivan Wyschnegradsky used the term ultra-chromatic for intervals smaller than the semitone and infra-chromatic for intervals larger than the semitone ; this same term has been used since 1934 by ethnomusicologist Victor Belaiev (Belyaev) in his studies of Azerbaijan and Turkish traditional music ( ; ). A similar term, subchromatic, has been used recently by theorist Marek Žabka . Ivor Darreg proposed (where and when?) the term xenharmonic. (See xenharmonic music). The Austrian composer Franz Richter Herf and his colleague at the Salzburg Mozarteum, the music theorist Rolf Maedel, preferred using the Greek word ekmelic when referring to "all the pitches lying outside the traditional twelve-tone system" . Some authors in Russia ( ; ; ; ; ; ), and some musicology dissertations ( ; ; ; ; , more references can be located here) disseminate the term ''микрохроматика'' (microchromatics), coined in the 1970s by Yuri Kholopov , to describe a kind of 'intervallic genus' (интервальный род) for all possible microtonal structures, both ancient (as enharmonic genus—γένος ἐναρμόνιον—of Greeks) and modern (as quarter-tone scales of Alois Haba); this generalization term allowed also to avoid derivatives such as микротональность (microtonality, which could be understood in Russian as a sub-tonality, which is subordinate to the dominating tonality, especially in the context of European music of the 19th century) and микротоника (microtonic, "a barely perceptible tonic"; see a clarification in ). Another Russian authors use more international adjective 'microtonal' and rendered it in Russian as 'микротоновый', but not 'microtonality' ('микротональность') ( ; ; ; etc. ). However are used 'микротональность' and 'микротоника' also. Some authors writing in French have adopted the term "micro-intervallique" to describe such music (e.g., ; ). Italian musicologist Luca Conti dedicated two his monographs to microtonalismo ( ; ), which is the usual term in Italian, and also in Spanish (e.g., as found in the title of ). The analogous English form, "microtonalism", is also found occasionally instead of "microtonality", e.g., "At the time when serialism and neoclassicism were still incipient a third movement emerged: microtonalism" . The term "macrotonal" has been used for intervals wider than twelve-tone equal temperament , or where there are "fewer than twelve notes per octave," though "this term is not very satisfactory and is used only because there seems to be no other" . The term "macrotonal" has also been used for musical form . History ) on E, divided into two tetrachords. ]] The Hellenic civilizations of ancient Greece left fragmentary records of their music—e.g. the Delphic Hymns. The ancient Greeks approached the creation of different musical intervals and modes by dividing and combining tetrachords, recognizing three genera of tetrachords: the enharmonic, the chromatic, and the diatonic. Ancient Greek intervals were of many different sizes, including microtones. The enharmonic genus in particular featured intervals of a distinctly "microtonal" nature, which were sometimes smaller than 50 cents, less than half of the contemporary Western semitone of 100 cents. In the ancient Greek enharmonic genus, the tetrachord contained a semitone of varying sizes (approximately 100 cents) divided into two equal intervals called dieses (single "diesis", δίεσις); in conjunction with a larger interval of roughly 400 cents, these intervals comprised the perfect fourth (approximately 498 cents, or the ratio of 4/3 in just intonation) . Theoretics usually described several diatonic and chromatic genera (some as chroai, "coloration" of one specific intervallic type), but the enarmonic genus was always the only one (argumented as one with the smallest intervals possible). Guillaume Costeley's "Chromatic Chanson", "Seigneur Dieu ta pitié" of 1558 used 1/3 comma meantone and explored the full compass of 19 pitches in the octave . The Italian Renaissance composer and theorist Nicola Vicentino (1511–1576) worked with microtonal intervals and built a keyboard with 36 keys to the octave known as the archicembalo. While theoretically an interpretation of ancient Greek tetrachordal theory, in effect Vicentino presented a circulating system of quarter-comma meantone, maintaining major thirds tuned in just intonation in all keys . In 1760 the French flautist Charles? De Lusse published a treatise, L'Art de la flute traversiere, all surviving copies of which conclude with a composition (possibly added a year or two after the actual publication of the volume) incorporating several quarter tones, titled Air à la grecque, accompanied by explanatory notes tying it to the realization of the Greek enharmonic genus and a chart of quarter-tone fingerings for the entire range of the one-keyed flute. Shortly afterward, in a letter published in the Mercure de France in September 1764, the celebrated flautist Pierre-Gabriel Buffardin mentioned this piece and expressed an interest in quarter tones for the flute ( ; ). Jacques Fromental Halévy composed a cantata "Prométhée enchaîné" for a solo voice, choir and orchestra (premiered in 1849), where in one movement (Choeur des Océanides) he used quartertones, to imitate the enharmonic genus of Greeks. In the 1910s and 1920s, quarter tones (24 equal pitches per octave) received attention from such composers as Charles Ives, Julián Carrillo, Alois Hába, Ivan Wyschnegradsky, and Mildred Couper. Alexander John Ellis, who in the 1880s produced a translation of Hermann Helmholtz's On the Sensations of Tone, proposed an elaborate set of exotic just intonation tunings and non-harmonic tunings . Ellis also studied the tunings of non-Western cultures and, in a report to the Royal Society, stated that they used neither equal divisions of the octave nor just intonation intervals . Ellis inspired Harry Partch immensely . During the Exposition Universelle of 1889, Claude Debussy heard a Balinese gamelan performance and was exposed to non-Western tunings and rhythms. Some scholars have ascribed Debussy's subsequent innovative use of the whole-tone (six equal pitches per octave) tuning in such compositions as the Fantaisie for piano and orchestra and the Toccata from the suite Pour le piano to his exposure to the Balinese gamelan at the Paris exposition , and have asserted his rebellion at this time "against the rule of equal temperament" and that the gamelan gave him "the confidence to embark (after the 1900 world exhibition) on his fully characteristic mature piano works, with their many bell- and gong-like sonorities and brilliant exploitation of the piano’s natural resonance" . Still others have argued that Debussy's works like L'isle joyeuse, La cathédrale engloutie, Prélude à l'après-midi d'un faune, La mer, Pagodes, Danseuses de Delphes, and Cloches à travers les feuilles are marked by a more basic interest in the microtonal intervals found between the higher members of the overtone series, under the influence of Helmholtz's writings . Emil Berliner's introduction of the phonograph in the 1890s allowed much non-Western music to be recorded and heard by Western composers, further spurring the use of non-12-equal tunings. Major microtonal composers of the 1920s and 1930s include Alois Hába (quarter tones, or 24 equal pitches per octave, and sixth tones), Julián Carrillo (24 equal, 36, 48, 60, 72, and 96 equal pitches to the octave embodied in a series of specially custom-built pianos), Ivan Wyschnegradsky (third tones, quarter tones, sixth tones and twelfth tones, non octaving scales) and the early works of Harry Partch (just intonation using frequencies at ratios of prime integers 3, 5, 7, and 11, their powers, and products of those numbers, from a central frequency of G-196) . Prominent microtonal composers or researchers of the 1940s and 1950s include Adriaan Daniel Fokker (31 equal tones per octave), Partch (continuing to build his handcrafted orchestra of microtonal just intonation instruments), and Eivind Groven. Digital synthesizers from the Yamaha TX81Z (1987) on and inexpensive software synthesizers have contributed to the ease and popularity of exploring microtonal music. Microtonality in electronic music Electronic music facilitates the use of any kind of microtonal tuning, and sidesteps the need to develop new notational systems . In 1954, Karlheinz Stockhausen built his electronic Studie II on an 81-step scale starting from 100 Hz with the interval of 51/25 between steps , and in Gesang der Jünglinge (1955–56) he used various scales, ranging from seven up to sixty equal divisions of the octave . In 1955, Ernst Krenek used 13 equal-tempered intervals per octave in his Whitsun oratorio, Spiritus intelligentiae, sanctus . In 1979–80 Easley Blackwood composed a set of Twelve Microtonal Etudes for Electronic Music Media, a cycle that explores all of the equal temperaments from 13 notes to the octave through 24 notes to the octave, including 15-ET and 19-ET }}. "The project," he wrote, "was to explore the tonal and modal behavior of all these equal tunings…, devise a notation for each tuning, and write a composition in each tuning to illustrate good chord progressions and the practical application of the notation" . In 1986, Wendy Carlos experimented with many microtonal systems including just intonation, using alternate tuning scales she invented for the album Beauty In the Beast. "This whole formal discovery came a few weeks after I had completed the album, Beauty in the Beast, which is wholly in new tunings and timbres" . Aphex Twin has been making microtonal electronic music with software and various analog and digital synthesizers since his 1994 album, Selected Ambient Works Volume II . In 2016, electronic music composed with arbitrary microtonal scales was explored on the album Radionics Radio: An Album of Musical Radionic Thought Frequencies by British composer Daniel Wilson, who derived his compositions' tunings from frequency-runs submitted by users of a custom-built web application replicating radionics-based electronic soundmaking equipment used by Oxford's De La Warr Laboratories in the late 1940s, thereby supposedly embodying thoughts and concepts within the tunings . Microtonality in rock music A form of microtone known as the blue note is an integral part of rock music and one of its predecessors, the blues. The blue notes, located on the third, fifth, and seventh notes of a diatonic major scale, are flattened by a variable microtone . Joe Monzo has made a microtonal analysis of the song Drunken-Hearted Man written and recorded by the delta blues musician Robert Johnson. Musicians such as Jon Catler have incorporated microtonal guitars like 31-tone equal tempered guitar and a 62-tone just intonation guitar in blues and jazz rock music . English rock band Radiohead has used microtonal string arrangements in its music, such as on "How to Disappear Completely" from the album Kid A . Australian band King Gizzard and the Lizard Wizard utilised custom microtonal guitars modified to play in 24-TET tuning and other microtonal instruments on their 2017 album Flying Microtonal Banana . American band Dollshot used quarter tones and other microtonal intervals in their 2018 album Lalande . See also * Sonido 13 * 3rd bridge * Arab tone system and maqam * Bohlen–Pierce scale * Continuum Fingerboard * Fokker periodicity blocks * Genus (music) * Harmony * Huygens-Fokker Foundation * Just intonation * Limit (music) * Microtuner * MIDI tuning standard * Music of India * Musical scale * Musical tuning * Partch's 43-tone scale * Quarter tone * Raga * Scala Western microtonal pioneers Pioneers of modern Western microtonal music include: * Henry Ward Poole (keyboard designs, 1825–1890) * Eugène Ysaÿe (Belgium, U.S.A., 1858–1931, used quarter tones in several of the Sonatas for Solo Violin, Op. 27) * Ferruccio Busoni (Italy, Germany, 1866–1924). Experimented with microtones, including third-tones. * Charles Ives (U.S.A., 1874–1954, quartertones) * Julián Carrillo (Mexico, 1875–1965) many different equal temperaments, look here or here (mostly Spanish but some English too) * Béla Bartók (Hungary, 1881–1945, rare uses of quartertones) * George Enescu (Romania, France, 1881–1955) (in Œdipe to suggest the enharmonic genus of ancient Greek music, and in the Third Violin Sonata, as inflections characteristic of Romanian folk music) * Karol Szymanowski (Poland, 1882–1937, used quartertones on the violin in Myths Op. 30, 1915) * Percy Grainger (Australia, 1882–1961, particularly works for his "free music machine") * Edgard Varèse (France, U.S.A., 1883–1965) * Mordecai Sandberg (Romania, Austria, Palestine, USA, Canada, 1897–1973) * Luigi Russolo (Italy, 1885–1947, used quartertones and eighth tones on the Intonarumori, noise instruments) * Mildred Couper (U.S.A., 1887–1974, quartertones) * Alois Hába (Czechoslovakia, 1893–1973, quartertones and other equal temperaments) * Ivan Wyschnegradsky (U.S.S.R. (Russia), France, 1893–1979, quartertones, twelfth tones and other equal temperaments) * Harry Partch (U.S.A., 1901–1974, just intonation, including a system of 43 unequal tones to the octave) * Eivind Groven (Norway, 1901–1977, 53ET) * Henk Badings (The Netherlands, 1907–1987, 31ET) * Maurice Ohana (France, 1913–1992, third tones (18-equal) temperament and quarter tones (24ET) most particularly) * Giacinto Scelsi (Italy, 1905–1988, intuitive linear tone deviations, quartertones, eighth tones) * Lou Harrison (U.S.A., 1917–2003, just intonation) * Ivor Darreg (U.S.A., 1917–1994) * Jean-Etienne Marie (France, 1919–1989, many different equal temperaments: 18ET, 24ET, 30ET, 36ET, 48ET, 96ET most particularly and polymicrotonality) * Franz Richter Herf (Austria, 1920–1989, 72-equal temperament, "ekmelic" music) * Iannis Xenakis (Greece, France, 1922–2001, quarter and third tones most particularly, occasionally eighth tones) * György Ligeti (Hungary, 1923–2006, Ramifications in quartertone tuning, natural harmonics in his Horn Trio, later just intonation in his solo concertos) * Luigi Nono (Italy, 1924–1990, quartetones, eighth tones and 16th tones) * Claude Ballif (France, 1924–2004, quartertones) * Tui St. George Tucker (1924–2004) * Pierre Boulez (France, 1925–2016) (first example of serial music with quartertones in his pieces Le Visage nuptial and Polyphonie X, but soon after abandoning microtonal elements) * Karlheinz Stockhausen (Germany, 1928–2007, in his electronic works many microtonal concepts, non-octaving scales in Studie II, just intonation in Gruppen and Stimmung, occasional microtonal instrumental and vocal writing throughout Licht) * Ben Johnston (U.S.A., b. 1926, extended just intonation) * Ezra Sims (U.S.A., b. 1928, 72-tone equal temperament) * Erv Wilson (1928–2016) * Carlton Gamer (U.S.A, born 1929, 7-tone, 19-tone, 22-tone, 31-tone equal temperament) * Alvin Lucier (U.S.A., b. 1931) * Joel Mandelbaum (U.S.A., b. 1932) * Krzysztof Penderecki (Poland, b. 1933, quartertones) * Easley Blackwood (b. 1933) * Alain Bancquart(France, b.1934) (quarter tones and 16th tones) * James Tenney (U.S.A., 1934–2006, just intonation, 72-tone equal temperament) * Terry Riley (U.S.A., b. 1935, just intonation) * La Monte Young (U.S.A., b. 1935, just intonation) * John Corigliano (U.S.A., b. 1938, quarter tones) * Douglas Leedy (b. 1938, just intonation, meantone) * Wendy Carlos (U.S.A., b. 1939, non-octaving scales) * Bruce Mather (Canada, b.1939, different equal temperaments, following Wyschnegradsky) * Brian Ferneyhough (Great Britain, b. 1943, quartertones, 31ET in Unity Capsule for solo flute,1976) * Joe Maneri (U.S.A., b. 1927) Recent Western microtonal composers * Clarence Barlow (b. 1945) * Gérard Grisey (1946–1998) (spectral approach to microintervals, quartertones, eighth tones) * Max Méreaux (b. 1946) * Tristan Murail (b. 1947) (spectral approach to microintervals, quartertones, eighth tones) * Glenn Branca (b. 1948) * Claude Vivier (1948–1983) * Dean Drummond (1949–2013) * Greg Schiemer (b. 1949) * Lasse Thoresen (b. 1949) * Warren Burt (b. 1949) * Manfred Stahnke (b. 1951) * James Erber (b. 1951) quarter tones * Rhys Chatham (b. 1952) * Kraig Grady (b. 1952) (invented acoustic instruments in just intonation & recurrent sequences) * David First (b. 1953) * Georg Friedrich Haas (b. 1953) * James Wood (b. 1953) * Pascale Criton (b. 1954) (different equal temperaments, most particularly very dense ETs such as the 96ET) * Paul Dirmeikis (b. 1954) * Stephen James Taylor (b. 1954) * Pascal Dusapin (b. 1955) (different equal temperaments, notably the 48ET) * Kyle Gann (b. 1955) * Johnny Reinhard (b. 1956) (different equal temperaments, just intonation, polymicrotonally) * Eric Mandat (b. 1957) * Erling Wold (b. 1958) * Michael Bach Bachtischa (b. 1958) * Lucio Garau (b. 1959) * Michael Harrison (b. 1959) (just intonation) * Martin Smolka (b. 1959) * Richard Barrett (b. 1959) * Georg Hajdu (b. 1960) * William Susman (b. 1960) * François Paris (b. 1961) * Franklin Cox (b. 1961) quarter tones, 1/12 tones, extended just intonation * Daniel James Wolf (b. 1961) * Claus-Steffen Mahnkopf (b. 1962) quarter tones, 1/8 tones * Harold Fortuin (b. 1964) * Marc Sabat (b. 1965) extended JI up to 23-limit * Georges Lentz (b. 1965) * Jeffrey Ching (b. 1965) (quartertones, ancient Chinese tunings, e.g. circle-of-fifths and just intonation) * Geoff Smith (b. 1966) * Trey Spruance (b. 1969) * Richard David James, aka Aphex Twin (b. 1971) * Paweł Mykietyn (b. 1971) * Yitzhak Yedid (b. 1971) * Yuri Landman (b. 1973) * Kristoffer Zegers (b. 1973) * Saman Samadi (b. 1984) * Taylor Brook (b. 1985) * Michael Waller (b. 1985) Western microtonal researchers * Mordecai Sandberg (1897–1973) * Christiaan Huygens (1629–1695) * Julián Carrillo (1875–1965) * Adriaan Daniël Fokker (1887–1972) * Ivan Wyschnegradsky (1893–1979) * Joseph Yasser (1893–1981) * Alois Hába (1893–1973) * Harry Partch (1901–1974) * Alain Daniélou (1907–1994) * Jean-Etienne Marie (1917–1989) * Erv Wilson (1928–2016) * Carlton Gamer (b. 1929) * Joel Mandelbaum (b. 1932) * James Tenney (1934–2006) * Clarence Barlow (b. 1945) * Gene Ward Smith (b. 1947) * Valeri Brainin (b. 1948) * Jacques Dudon (b. 1951) * William Sethares (b. 1955) * Georg Hajdu (b. 1960) * Bob Gilmore (1961–2015) * Marc Sabat (b. 1965) References * |reference=Adèr, Lidiâ Olegovna Лидия Олеговна. 2011a. 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Schweizer Jahrbuch für Musikwissenschaft/Annales suisses de musicologie/Annuario svizzero di musicologia 22, edited by Joseph Willimann. Bern: Verlag Peter Lang AG. .}} * |reference=Barbieri, Patrizio. 2003. "Temperaments, Historical". In Piano: An Encyclopedia, second edition, edited by Robert Palmieri and Margaret W. Palmieri, . New York: Routledge. *Barbieri, Patrizio. 2008. Enharmonic instruments and music, 1470-1900. Latina: Il Levante Libreria Editrice. .}} * |reference=Barbieri, Patrizio, Alessandro Barca, and conte Giordano Riccati. 1987. Acustica accordatura e temperamento nell'illuminismo Veneto. Pubblicazioni del Corso superiore di paleografia e semiografia musicale dall'umanesimo al barocco, Serie I: Studi e testi 5; Pubblicazioni del Corso superiore di paleografia e semiografia musicale dall'umanesimo al barocco, Documenti 2. Rome: Edizioni Torre d'Orfeo.}} * |reference=Barbieri, Patrizio, and Lindoro Massimo del Duca. 2001. 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Vanessa Carr’s website (29 February). (Accessed 2 April 2009)}} * |reference=Colonna, Fabio. 1618. La sambuca lincea, overo Dell'istromento musico perfetto. Naples: C. Vitale. Facsimile reprint of a copy containing manuscript critical annotations by Scipione Stella (1618–1624), with an introduction by Patrizio Barbieri. Musurgiana 24. Lucca, Italy: Libreria Musicale Italiana, 1991.}} * |reference=Daniels, Arthur Michael. 1965. "Microtonality and Mean-Tone Temperament in the Harmonic System of Francisco Salinas". Journal of Music Theory 9, no. 1 (Spring): 2–51.}} * |reference=Dumbrill, Richard J. 2000. The Musicology and Organology of the Ancient Near East, second edition. London: Tadema Press. .}} * |reference=Fink, Robert. 1988. "The Oldest Song in the World". Archaeologia Musicalis 2, no. 2:98–100.}} * |reference=Fritsch, Johannes G. 2007. "Allgemeine Harmonik, Tonsysteme, Mikrotonalität: Ein geschichtlicher Überblick". In Orientierungen: Wege im Pluralismus der Gegenwartsmusik, edited by Jörn Peter Hiekel, 107–22. Veröffentlichungen des Instituts für Neue Musik und Musikerziehung Darmstadt 47. Mainz: Schott Musik International. .}} * |reference=Gilmore, Bob. 1998. Harry Partch: A Biography. New Haven: Yale University Press. .}} * |reference=Haas, Georg Friedrich. 2007. "Mikrotonalität und spektrale Musik seit 1980". In Orientierungen: Wege im Pluralismus der Gegenwartsmusik, edited by Jörn Peter Hiekel, 123–29. Veröffentlichungen des Instituts für Neue Musik und Musikerziehung Darmstadt 47. Mainz: Schott Musik International. .}} * |reference=Hába, Alois. 1927. Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems. Leipzig: Kistner & Siegel.}} * |reference=Jedrzejewski, Franck. 2003. Dictionnaire des musiques microtonales of Microtonal Musics. Paris: L'Harmattan. .}} * |reference=Jedrzejewski, Franck. 2014. Dictionnaire des musiques microtonales. Nouvelle édition revue et augmentée (1892–2013) of Microtonal Musics. Paris: L'Harmattan. .}} * |reference=Johnston, Ben. 2006. 'Maximum Clarity' and other writings on music, ed. B. Gilmore. Urbana and Chicago: University of Illinois Press.}} * |reference=Kotschy, Johannes. 2008. "Mikrotonalität: Eine Zeiterscheinung?" Österreichische Musikzeitschrift 63, no. 7 (July): 8–15.}} * |reference=Landman, Yuri. 2008. "Third Bridge Helix: From Experimental Punk to Ancient Chinese Music and the Universal Physical Laws of Consonance". Perfect Sound Forever (online music magazine) (accessed 6 December 2008).}} * |reference=Landman, Yuri. n.d. "Yuichi Onoue’s Kaisatsuko" on Hypercustom.com (accessed 31 March 2009).}} * |reference=Leedy, Douglas. 1991. "A Venerable Temperament Rediscovered". Perspectives of New Music 29, no. 2 (Summer): 202–11.}} * |reference=Mandelbaum, M. Joel. 1961. "Multiple Division Of the Octave and the Tonal Resources of the 19 Tone Temperament". Ph.D. thesis. Bloomington: Indiana University.}} * |reference=Mosch, Ulrich. 2008. "Ultrachromatik und Mikrotonalität: Hans Zenders Grundlegung einer neuen Harmonik". In Hans Zender: Vielstimmig in sich, edited by Werner Grünzweig, Anouk Jeschke, and Jörn Peter Hiekel, 61–76. Archive zur Musik des 20. Jahrhunderts, No. 12. Hofheim:Wolke. .}} * |reference=Stahnke, Manfred. 2010. "About Backyards and Limbos: Microtonality Revisited". In Concepts, Experiments and Fieldwork: Studies in Systematic Musicology and Ethnomusicology, edited by Rolf Bader, Christiane Neuhaus, and Ulrich Morgenstern, with a prefaceby Achim Reichel, 297–314. Frankfurt am Main and New York: Peter Lang. .}} * |reference=Vitale, Raoul. 1982. "La Musique suméro-accadienne: gamme et notation musicale". Ugarit-Forschungen 14: 241–63.}} * |reference=Werntz, Julia. 2001. "Adding Pitches: Some New Thoughts, Ten Years after Perspectives of New Music's 'Forum: Microtonality Today'". Perspectives of New Music 39, no. 2 (Summer): 159–210.}} * |reference=Wood, James. 1986. "Microtonality: Aesthetics and Practicality". The Musical Times 127, no. 1719 (June): 328–30.}} * |reference=Zweifel, Paul. 1996. "Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach". Perspectives of New Music 34, no. 1 (Winter): 140–61.}} External links * * Aikin, Jim. 2003. Jim Aikin's article on alternative tuning in electronic music * Anon. n.d.. "Nicola Vicentino (1511–1576)". IVO: Sacred Music in the Italian Cinquecento outside Venice and Rome, edited by Chris Whent. Here Of A Sunday Morning website. (Accessed 19 August 2008) * Chalmers, John. Dr. John Chalmers Divisions of the Tetrachord * Loli, Charles. 2008. " Microtonalismo". (Article on alternative tuning in Peruvian music) * Solís Winkler, Ernesto. 2004. "Julián Carrillo and the 13th Sound: A Microtonal Musical System". (Accessed 19 August 2008) *Wilson, Erv. "Wilson Archives of papers on microtonal theory" * Links to microtonal composers at Xenharmonic Wiki * Links to microtonal projects around the world at Xenharmonic Wiki * Offtonic Microtonal Synthesizer, a browser-based synth to explore microtonal tunings with a QWERTY keyboard Category:Microtonality Category:Ancient Greek music Category:Post-tonal music theory